A new invariant for knotted graphs defined by label bracket.
problem Defining an invariant for knotted trivalent graphs.
method Generalizing Akimova and Manturov's construction to define the label bracket.
result The label bracket defines an isotopy invariant of knotted trivalent graphs.
We propose to generalize the volume conjecture to knotted trivalent graphs and we prove the conjecture for all augmented knotted trivalent graphs. As a corollary we find that for any link L there is a link containing L for which the volume conjecture holds.
New signatures for knotted graphs linked to classical knot signatures.
problem Defining invariants for knotted trivalent graphs.
method Using branched covers to define and relate new signatures to classical knot signatures.
result Computable invariants for Kinoshita's knotted theta graph.
Enhanced trivalent tangles and handlebody-tangles invariants created.
problem Creating invariants for trivalent and handlebody-tangles.
method Using enhanced trivalent tangles and classical knot theory.
result Constructed invariants for trivalent and handlebody-tangles.
The paper classifies virtual knot polynomials and trivalent graph invariants using skein theory.
problem Classifying virtual knot polynomials and trivalent graph invariants with specific conditions.
method Skein-theoretic techniques applied to classify invariants with smallness conditions.
result Classification of all non-trivial invariants of trivalent graphs and skein theories of virtual tangles.
Minimal sets of moves for isotopic knots and trivalent graphs identified.
problem Identifying minimal sets of moves for isotopic knots and trivalent graphs.
method Provided and proved the existence of minimal generating sets of oriented Reidemeister moves for isotopic knots and spatial trivalent graphs.
result Twelve minimal generating sets of oriented Reidemeister moves for isotopic knots and ten for spatial trivalent graphs identified.
In 1965, E. C. Zeeman proved that the (+/-)-twist spin of any knotted sphere in (n-1)-space is unknotted in the n-sphere. In 1991, Y. Marumoto and Y. Nakanishi gave an alternate proof of Zeeman's theorem by using the moving picture method. In this paper, we define a knotted 2-dimensional foam which is a generalization …
The paper generalizes virtual knot theory using multiple types of virtual crossings.
problem Generalizing virtual knot theory to include multiple types of virtual crossings.
method Starting with graph theory, the paper reviews previous work and then constructs multi-virtual knots and links.
result The multiplicity of virtual crossings allows for a broader application of the Penrose evaluation to all trivalent graphs.
We construct an extension of the Kontsevich integral of knots to knotted trivalent graphs, which commutes with orientation switches, edge deletions, edge unzips, and connected sums. In 1997 Murakami and Ohtsuki [MO] first constructed such an extension, building on Drinfel'd's theory of associators. We construct a step …
In this paper we show that via the configuration space integral construction a non-trivalent graph cocycle can also yield a non-zero cohomology class of the space of higher (and even) codimensional long knots. This simultaneously proves that the Browder operation induced by the operad action defined by R. Budney is not…
New algebraic structures help color handlebody-knots for invariants.
problem Coloring handlebody-knots for invariants.
method Introducing G-families of biquandles, partially multiplicative biquandles, and group decomposable biquandles. result Enhanced polynomial invariant of handlebody-knots using group G. Census of hyperbolic Platonic manifolds and their complements.
problem Classifying hyperbolic Platonic manifolds and their complements.
method Generalized earlier work on ideal tetrahedra to octahedra, identifying complements of augmented knotted trivalent graphs.
result Identification of complements of augmented knotted trivalent graphs in hyperbolic Platonic manifolds.
Knotted trivalent graphs (KTGs) form a rich algebra with a few simple operations: connected sum, unzip, and bubbling. With these operations, KTGs are generated by the unknotted tetrahedron and Moebius strips. Many previously known representations of knots, including knot diagrams and non-associative tangles, can be tur…
It had been known since old times [MO, Da] that there exists a universal finite type invariant ("an expansion") Z^{old} for Knotted Trivalent Graphs (KTGs), and that it can be chosen to intertwine between some of the standard operations on KTGs and their chord-diagrammatic counterparts (so that relative to those operat…
The Kontsevich integral of a knot is a powerful invariant which takes values in an algebra of trivalent graphs with legs. Given a Lie algebra, the Kontsevich integral determines an invariant of knots (the so-called colored Jones function) with values in the symmetric algebra of the Lie algebra. Recently A. Kricker and …
This article addresses the two significant aspects of Ozsváth and Szabó's knot Floer cube of resolutions that differentiate it from Khovanov and Rozansky's HOMFLY-PT chain complex: (1) the use of twisted coefficients and (2) the appearance of a mysterious non-local ideal. Our goal is to facilitate progress on Rasmussen…
Study examines how changing regions affects planar graphs.
problem Effect of region crossing change on planar trivalent graphs.
method Investigation of region crossing changes on planar trivalent graphs.
result Effect of region crossing change on planar trivalent graphs.
A proof of a shadow formula for a specific invariant using skein theory.
problem Calculating SU(2)-Reshetikhin-Turaev-Witten invariants for 3-manifolds and graphs.
method Skein theory
result A short proof of Turaev's shadow formula for the specified invariants.
Introduces Niebrzydowski algebras for trivalent spatial graphs and handles.
problem Counting and distinguishing trivalent spatial graphs and handlebody-links.
method Defines Niebrzydowski algebras with ternary operation and partially defined multiplication, motivated by Reidemeister moves.
result Niebrzydowski algebras can distinguish some trivalent spatial graphs and handlebody-links.
New invariants detect knots and add under operations.
problem Detecting knots and links in 3-manifolds.
method Defining two new families of invariants related to bridge and tunnel numbers.
result The invariants detect the unknot and are additive under specific operations.
Proves Markov theorem for trivalent braids using L-move approach.
problem Proving Markov theorem for trivalent braids.
method Follows L-move approach to prove Markov theorem.
result Proves one-move Markov-type theorem and algebraic Markov-type theorem for trivalent braids.
We construct a state model for the two-variable Kauffman polynomial using planar trivalent graphs. We also use this model to obtain a polynomial invariant for a certain type of trivalent graphs embedded in three-dimensional space.
Invariants for trivalent graphs using algebraic colorings.
problem Creating an invariant for virtual trivalent spatial graphs.
method Colorings by virtual Niebrzydowski algebras.
result Generalization and computational implementation of invariants.
New thin position for graphs in 3-manifolds combines ideas from knot theory and 3-manifold topology.
problem Defining a new thin position for graphs in 3-manifolds.
method Combining thin position concepts from knot theory and 3-manifold topology.
result Defines new invariants of knots, links, and graphs in 3-manifolds.
We associate a moduli problem to a colored trivalent graph; such graphs, when planar, appear in the state-sum description of the quantum sl(N) knot polynomial due to Murakami, Ohtsuki, and Yamada. We discuss how the resulting moduli space can be thought of a representation variety. We show that the Euler characteristic…
We give a description of local and global moves on a class of locally planar trivalent graphs and we show that it contains λ-Scale calculus, therefore in particular untyped lambda calculus. Surprisingly, the beta reduction rule comes from a local "sewing" transformation of trivalent locally planar graphs.
Study combinatorial analogues of Kronheimer-Mrowka theory for graphs.
problem No specific problem stated; focuses on theory development.
method Introduce combinatorial equivariant analogues of Kronheimer-Mrowka homology theory.
result Developed combinatorial analogues for planar trivalent graphs.
New method to classify simple Smale flows on S3.
problem Classifying simple Smale flows on S3. method Embedded template and Kauffman's invariant of spatial graphs.
result Isotopic classification of simple Smale flows on S3. Motivated by a possible connection between the SU(N) instanton knot Floer homology of Kronheimer and Mrowka and sl(N) Khovanov-Rozansky homology, Lobb and Zentner recently introduced a moduli problem associated to colourings of trivalent graphs of the kind considered by Murakami, Ohtsuki and Yam…
Kronheimer-Mrowka's instanton homology dimension equals Tait colorings.
problem Calculating the dimension of a specific homology group for plane trivalent graphs.
method Using SO(3) instanton Floer homology, the dimension is shown to be equal to the number of Tait colorings.
result The dimension of J#(G) is equal to the number of Tait colorings of G.
Constructs integer-valued cohomology classes from graph cocycles.
problem Constructing nontrivial cohomology classes from graph cocycles.
method Gluing compactified configuration spaces to construct integer-valued classes from integer-valued graph cocycles.
result Obtains nontrivial classes from trivalent graph cocycles.
Classifies 1-connected 2-stratifolds using graph theory.
problem Classifying 1-connected trivalent 2-stratifolds.
method Uses associated labeled graphs and operations to construct all graphs representing 1-connected 2-stratifolds.
result Developed methods to construct all graphs representing 1-connected 2-stratifolds from a single vertex.
The paper extends foam theory to more complex trivalent graphs.
problem Extending foam theory to more complex trivalent graphs.
method Considering foams with singular vertices homeomorphic to cones over more general planar trivalent graphs.
result Modules associated with the dodecahedron graph are free of rank 60.
Homology theory for qualgebras yields knotted graph and foam invariants.
problem Developing a homology theory for qualgebras.
method Constructing a classifying space from prisms and adding degenerate cells.
result Homotopy classes of maps from spheres to the classifying space of G. The dual to a tetrahedron consists of a single vertex at which four edges and six faces are incident. Along each edge, three faces converge. A 2-foam is a compact topological space such that each point has a neighborhood homeomorphic to a neighborhood of that complex. Knotted foams in 4-dimensional space are to knotted…
This is a substantially revised version. The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots. We introduce a second grading of the Kontsevich integral, the Eule…
Proves Alexander- and Markov-type theorems for virtual trivalent braids.
problem Classifying virtual trivalent braids and graphs.
method Two versions of the Markov-type theorem: algebraic and L-move based.
result Established new theorems for virtual trivalent braids.
Paper defines a polynomial invariant for surface-links using quantum A_2 invariant.
problem Defining a polynomial invariant for surface-links.
method Using the quantum A_2 invariant and Yoshikawa moves, a polynomial is defined for marked graph diagrams.
result The polynomial invariant is useful for studying ribbon 2-knots.
New method calculates knot and link biquandle brackets using trace diagrams.
problem Computing biquandle brackets of knots and links efficiently.
method Using trace diagrams to compute biquandle brackets of oriented knots and links.
result Identified algebraic conditions for strand moves and stop conditions.
The paper refines 2-factor homology to a stable homotopy type for planar trivalent graphs with perfect matchings.
problem Developing a stable homotopy type for planar trivalent graphs with perfect matchings.
method Defining a cover functor from the 2-factor flow category to the cube flow category, realizing the 2-factor spectrum, and showing it's an invariant.
result The stable homotopy type of the 2-factor spectrum is an invariant of planar trivalent graphs with perfect matchings.
The paper studies tunnel and bridge numbers of composite genus 2 spatial graphs.
problem Understanding the tunnel and bridge numbers of composite genus 2 spatial graphs.
method Analyzes connected sum and trivalent vertex sum operations on genus 2 spatial graphs, proving bounds for tunnel and bridge numbers.
result Sharp bounds for the tunnel number of composite genus 2 spatial graphs, including lower bounds for bridge numbers.
New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.
problem Defining invariants for framed 3-manifolds with semi-simple Lie groups.
method Constructing graph complexes and cocycles, including self-loops, to define invariants.
result Higher-loop invariants can be defined by graph cocycles with or without self-loops.
New algebras and maps defined in knot Floer homology for trivalent vertices.
problem Categorification of knot Floer homology for trivalent vertices.
method Definition of new algebras, local bimodules, and bimodule maps in bordered knot Floer homology.
result Categorification of representations of U_q(gl(1|1)^-).
The paper explores new quandle systems for handlebody-links and spatial graphs.
problem Developing new invariants for handlebody-links and spatial graphs.
method Investigates and constructs new algebraic systems (quandles) to generalize existing ones.
result Provides necessary conditions for colouring invariants of knotted handlebodies.
Let Gg,b be the set of all uni/trivalent graphs representing the combinatorial structures of pant decompositions of the oriented surface of genus g with b boundary components. We describe the set Ag,b of all automorphisms of graphs in Gg,b showing that, up to suitable moves changing the graph within …
The generalized volume conjecture and the AJ conjecture (a.k.a. the quantum volume conjecture) are extended to $U_q(\fraksl_2)$ colored quantum invariants of the theta and tetrahedron graph. The $\SL(2,\bC)$ character variety of the fundamental group of the complement of a trivalent graph with E edges in S3 is a L…
Python code constructs connected 2-stratifolds from graphs.
problem Creating models of connected trivalent 2-stratifolds.
method Developed operations on associated labeled graphs to construct connected 2-stratifolds.
result Implemented Python code to automate the construction process.
The paper extends graph signatures to Klein graphs and foams, linking signatures to knot properties.
problem Extending graph signatures to Klein graphs and foams.
method Developed an analogy of Murasugi's bounds and used signatures to lower bound knot properties.
result Lower bounds on negative orbifold Euler characteristics and unknotting numbers.